Definition:Set of Integer Multiples

Definition

The set $n \Z$ is defined as:

$\set {x \in \Z: n \divides x}$

for some $n \in \Z_{>0}$.

That is, it is the set of all integers which are divisible by $n$, that is, the set of integer multiples of $n$.

Thus we have:

$n \Z = \set {\ldots, -3 n, -2 n, -n, 0, n, 2 n, 3 n, \ldots}$

Also see

• Definition:Principal Ideal of Ring

• Results about sets of integer multiples can be found here.

Examples

Set of Even Numbers

The set of even integers is defined as:

$2 \Z := \set {x \in \Z: 2 \divides x}$

That is:

$2 \Z := \set {\ldots, -6, -4, -2, 0, 2, 4, 6, \ldots}$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.2$. Subgroups: Example $92$